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In photorefractive waveguide arrays, the process and extent of spectral broadening of femtosecond laser pulse caused by self-phase modulation are studied theoretically and experimentally. The threshold of self-phase modulation is more than two times larger than the common threshold in a bulk sample, which affects the extent of spectral broadening dramatically. The coupling length and the ratio between the common threshold and the input peak intensity of the femtosecond laser pulse are the two key parameters dominating these phenomena. The experimental results confirm the theoretical expectation.
©2010 Optical Society of America
1. Introduction
In recent years, light propagation in waveguide arrays (a discrete system) has aroused a great deal of interest. For instance, in fabricated or optically-induced (photorefractive) photonic lattices, the evanescent coupling among adjacent waveguides has enabled the study of a host of intriguing phenomena such as Anderson localization [1], Bloch oscillations [2], Zener tunneling [3], Rabi oscillation [4], and quantum walks [5]. Waveguide arrays introduce a band gap in spatial frequency domain or wave-vector space, which leads to management of beam diffraction and anomalous refraction and diffraction [6,7]. In nonlinear regime, a variety of localized states including discrete solitons [8,9], gap solitons [10,11], discrete surface solitons [12,13], and discrete vortex solitons [14,15] has been demonstrated. Soliton-based all-optical switching schemes have also been suggested [16–18].
Introducing femtosecond laser pulse into waveguide arrays is important for understanding nonlinear effects driven by high peak power. It also brings about many new phenomena in time-domain, such as spatiotemporal X wave [19], pulse compression [20] and reshaping [21] in AlGaAs waveguides, spatiotemporal localization in silica waveguide arrays [22], spatiotemporal soliton in silicon-on-insulator photonic wires arrays [23,24]. But in the research of photorefractive waveguide arrays, there are unexplored subjects in time-domain where a femtoseond laser pulse propagating through the waveguide arrays is considered, in spite of a great mount of work done in spatial domain. Here we demonstrated nonlinear spectrum broadening of femtosecond laser pulses in optically-induced (photorefractive) waveguide arrays.
When the laser pulses propagate through waveguide arrays, they are temporally reshaped by the effects of dispersion and self-phase modulation (SPM). The SPM effect is typical dominant and could lead to some interesting nonlinear consequences, including, for example, pulse shortening accompanied with dispersion compensation [20] under two appropriate conditions. One is that the peak intensity exceeds a value that is high enough to make the third-order susceptibility to play the role, commonly known as the threshold of SPM [25]. The other is that, for a sample with a certain length, the input peak intensity must exceed a value that is high enough to make the nonlinear length LNL (defined in section 2) comparable to the effective propagation length Leff. This threshold is induced not only by the material but also by the waveguide array structures. When the input intensity is beyond these two thresholds (the second threshold is normally larger than the first one), the refractive index change induced by the light will introduce an additional nonlinear phase shift during propagation. In this case, the spectral broadening can be found as long as the propagation length is long enough to accumulate the maximum phase shift (phase shift of the peak intensity component of a pulse [26–29]) to around 1 radian.
In this paper, we report our study of nonlinear spectral broadening of femtosecond laser pulses in a photorefractive waveguide arrays fabricated in an iron-doped lithium niobate crystal. Comparing the nonlinear process in a common diffraction case (named as “Case I” in this paper) with a linearly discrete diffraction case (named as “Case II” in this paper), we discuss the threshold of SPM and spectrum broadening in waveguide arrays. The array threshold, which is the second threshold of SPM in waveguide arrays from coupling-induced intensity variation and the nonlinear spectrum broadening induced by femtosecond laser pulses, is found to be more than two times larger than the common threshold in the bulk sample.
2. Fundamentals
2.1 The bandwidth broadening accumulation induced by SPM
SPM-introduced bandwidth broadening amount can be estimated according to the maximum phase shift at the pulse center [26–28]:
where the nonlinear length LNL is defined as which is an important coefficient of the nonlinearity magnitude; P0 is the peak power of the laser pulse, and γ is the nonlinear parameter of a waveguide, defined aswhere is the nonlinearity coefficient of the material, defined by the equation c is the optical velocity in vacuum, is the central angular frequency of the pulse, and is the effective area of the beam propagating in a sample. In waveguide arrays, because the beam intensity varies during propagation, the effective length Leff will be changed with respect to the intensity I and coupling coefficient LC, i.e. Leff = Leff (I, LC).Substituting γin Eq. (1) using Eq. (2) and considering Leff in waveguide arrays, we get
where2.2 The intensity variation in waveguide arrays
For convenience, let us consider the simplest discrete diffraction case: a single mode linear diffraction in a weakly coupled waveguide array (the coupling only occurs between adjacent channels) [30].
where En indicates the electric field amplitude in the nth channel, C is the coupling constant between adjacent channels and C = π / (2LC), and the beam propagates along the z axis. The analytic solution of the amplitude of the nth channel can be expressed by a Bessel Function [30]:2.3 Accumulation of spectrum broadening in case II
Discrete diffraction induces the beam intensity variation in each waveguide channel owing to coupling between adjacent waveguides. In some cases, the intensities in some channels are lower than the first (common) threshold of SPM effect and these components are actually noncontributive. That is, the intensity variation will shorten the effective propagation length Leff and induce the second threshold although the accumulation of maximum phase shift is linearly integrated in one waveguide channel.
To study the nonlinear process of the discrete diffraction, we trace the ϕmax accumulation process of the whole beam. Because all the phase shifts of the pulses in the channels contribute to the total phase shift of the whole beam, all the contributive components should be taken into consideration to calculate the maximum phase change in Case II. It is not a right way to simply summarize the nonlinear phase shifts of all the pulses to calculate that of the whole output beam. Considering N pulses in waveguide arrays, whose nonlinear phase shift are all π, the phase shift of the whole beam must be π, rather than Nπ. The contribution of each pulse relates to its peak intensity. So both the phase shift of each pulse and its intensity proportion to that of the whole beam must be considered. Under the condition of our experiment, we can write out a semi-empirical formula for weight factor: Ii /I0, where Ii is the peak intensity of the pulse in the i th channel.
Considering all the phase shifts of the pulses in the channels that contribute to the total phase shift of the whole beam and the weight factor, the maximum phase change of the whole output beam in Case II is:
where λ0 is the central wavelength of the input beam, n is the number of waveguides, L is the length of the sample, In(z,Ith)/I0 is the weight of the component In(z,Ith) of the whole intensity, andwhere In(z) is the peak intensity in channel n at propagation length z.Figure 1 is the simulation results of the maximum phase shift in Case I (dot) and in Case II (solid) in a LiNbO3: Fe crystal (1.4 cm length). The parameters are in accordance with the experiment (see section 3). It is obvious that the phase shift doesn’t accumulate until the input peak intensities exceed the threshold of SPM (inset in Fig. 1, before the fork). Above the threshold, the nonlinear phase shift begins to be accumulated during propagation, but the effective lengths in Case II (solid) are different from that in Case I (dot). As the input energy increases, the bandwidth increases nonlinearly with respect to input power.
Fig. 1 Nonlinear phase shift with respect to input peak intensity in Case I (dot) and in Case II (solid), respectively. The inset is the enlargement of the beginning of the curves.
In Fig. 2 , we have simulated the maximum phase shift in the waveguide arrays with different normalized coupling length LC /L (a) or with different ratio η = Ith /I0 (b). In the graph (a), we can see that the maximum phase shift will accumulate in a same progress with a certain ratio of η = 23.7%. In this graph, the maximum phase shift accumulation occurs only in the first coupling length. This is because the beam intensity will fall below the threshold of SPM after the first coupling. Although the coupling is very strong (Fig. 2 (a), dash-dot), there is also an effective length for bandwidth broadening in the first coupling length (Fig. 2 (a), inset graph), but the effective length will be determined by the coupling length and the sample length. In the same way, the ratio η affects the maximum phase shift (Fig. 2 (b)) in the accumulating progress. Consequently, there are two decisive factors, the normalized coupling length LC /L and the ratio η, which affect the process and amount of bandwidth broadening.
Fig. 2 Φmax accumulation during propagation (sample length L = 1.4 cm). (a): Φmax according to different coupling length (normalized by sample length L) for η = Ith /I0 = 23.7%: LC /L = 4.62 × 10−5 (dash-dot), 0.54 (solid), and 76.92 (dash), respectively; The inset is the enlargement of the beginning of solid and dash-dot curves. (b): Φmax for LC /L = 0.11 according to the ratio of threshold to input intensity η = Ith /I0 = a: 11.41%, b: 13.69%, c: 19.56%, d: 32.59%, and e: 1369%, respectively.
Based on these results, a conclusion can be drawn that the normalized coupling length determines the effective propagation length Leff, and the ratio η determines the accumulating process of Φmax during propagation.
3. Experimental setup
The experimental setup is shown in Fig. 3 . A one-dimensional waveguide arrays was fabricated with period Λ ≈11 µm (inset) by photorefractive effect in a LiNbO3: Fe crystal (0.05 wt.% Fe, 1.4 cm propagation length of the sample). An extraordinarily polarized beam produced by a YAG: Nd laser (with 532 nm) was expanded by a telescope (lens L1 and L2), and then split into two beams by a beam splitter. The two beams interfered in the sample with a cross angle of 2.8 degree in air. For an intensity of 100 mW/cm2 (intensity of each beam), the waveguide arrays was built up after about 90 seconds. The femtosecond (fs) laser pulse (800 nm, 120 fs, 1 kHz) from a regenerative amplifier (from Spectra-Physics Corp.) was introduced into one channel by a cylindrical lens (CL1). The output pattern is captured by a CCD camera after lens L3 and an objective mounted on the camera. A fiber spectrometer picks up the spectrum of the output beam from all the channels reflected by a beam splitter at the focal point of L3.
Fig. 3 Experimental setup. The setup is composed of three parts: waveguide arrays writing system (I), femtosecond laser propagation system (II), and acquisition and analysis system (III). The wave vector of the waveguide arrays fabricated in the sample is along the optical axis of the crystal (inset). P1-P2: polarizer, L1-L3: focal lens, BS: beam splitter, CL1: cylindrical lens, YAG: Nd laser, Spitfire: femtosecond pulse laser, Spectrometer: fiber spectrometer (Ocean Optics Inc., HR4000CG-UV-NIR Spectrometer), CCD (DataRay Inc., WinCamD-UCM) mounted a 4 × objective.
4. Results and discussion
The output pattern after 1.4 cm propagation has been simulated for the Case II (Fig. 4(b) ), and it is consistent with the experimental result (Fig. 4(c)). According to above simulation of the coupling procedure and the numerical simulation result of the intensity variation (Fig. 4(a)), the maximum phase shift accumulation process during discrete diffraction was calculated for an input peak intensity of 193GW/cm2 and the result is shown in Fig. 4(d). As we mentioned above, it is different from bulk sample and induces different threshold and process of spectrum broadening.
Fig. 4 Simulation of discrete diffraction and the nonlinear phase change versus propagation length. Upper: simulation of discrete diffraction propagation (left) and the output pattern (middle), and the experimental output pattern (right); Lower, simulated maximum phase change Φmax versus propagation length.
Figure 5 shows the discrete diffraction pattern of femtosecond laser pulse at output facet of the sample. Shown in 3D image Fig. 5(a), it is clear that much of the power is coupled into bilateral channels. Averaging the output power along Y coordinate in Fig. 5(a), we get a plot curve of the output power for the channels. Figure 5(b) is the plot curves for the different input intensities in our experiment and demonstrates that the discrete diffraction pattern maintains well, when the input intensity is increased form 1.18GW/cm2 to 824GW/cm2. Beyond the highest input intensity in our experiment, we don’t get soliton-like phenomena until the photorefractive waveguide arrays are destroyed.
Fig. 5 Discrete diffraction pattern of femtosecond laser pulse at output facet. (a): 3D image of discrete diffraction; (b): power distribution of discrete diffraction for different input peak intensities, which has been averaged along Y coordinate.
Figure 6 shows the output spectra captured at different input peak intensities in Case II. In the experiment, 120 groups of output spectra for different input peak intensities for case I and case II were taken, respectively. For each input intensity, the experiment was repeated for 5~31 times to eliminate the false phenomena and wrong judgment. For a clear comparing result, we only show 4 and 7 pairs of spectra as the representative data in Fig. 6 and Fig. 7 . It shows that the threshold in Case II is around 193 ± 2GW/cm2 (the error value is the maximum error in the experiment, and the same for Case I), which is about 2.1 times larger than that in Case I (92.1 ± 0.9GW/cm2, which was acquired from the same setup using the unchanged sample without waveguide arrays). It confirms our theoretical expectation, which can also be clearly seen in Fig. 7.
Fig. 6 Output Spectra in Case II at an input intensity of 1.18, 193, 438 and 824 GW/cm2 for (a)-(d), respectively.
Fig. 7 Contrast of the output spectra for Case I (a) and case II (b) versus the input peak intensities.
The oscillation structure in the spectral distribution indicates an SPM-induced nonlinear chirp, which is in agreement with theoretical prospect [27]. In Fig. 6, we can also see that the broadened spectrum is a little asymmetry. That is because of the self-steepening and the third-order dispersion, which can be simulated by Eq. (4).3.1) in Ref [27]. But the main determination of the broadening behavior is the waveguide arrays structure, which controls spectrum broadening.
Figure 7 shows the contrast of the output spectra at different input peak intensities in Case I (a) and in Case II (b). As expected in section 2, for the reason that the power is distributed into channels, the output spectra of (a) are wider than that of (b) and the threshold is opposite. But if we change the propagation length and the coupling length to be a proper value, the spectrum broadening in case II can also be wider than that in Case I, because the power is concentrated into channel for a relatively long distance [31].
Furthermore, since the nonlinearity can be controlled by structures for a given sample, the dispersion and the nonlinear chirp can be balanced by waveguide arrays to build a temporal soliton. In this case, the nonlinear process will occur strongly at the first part of propagation length, and the two effects can only be balanced at output face with a certain propagation length. It can be regarded as a kind of new higher order temporal soliton with asymmetric temporal process.
5. Summary
We have fabricated a waveguide array of 11 µm period in an iron doped lithium niobate crystal and injected a 120 fs pulse laser into one channel waveguide to study its discrete diffraction. We discuss the threshold of SPM and spectrum broadening in photorafractive waveguide arrays. The array threshold was found to be about more than two times larger than the common threshold in the bulk sample because of the intensity variation in the waveguide channels induced by the coupling between them. It also changes the maximum phase shift accumulation and then the bandwidth broadening. In this way, the waveguide arrays can effectively control the spectrum broadening by the coupling length and the ratio between the bulk threshold and the array threshold. If the propagation length and the coupling length are set to a proper value, the array threshold can also be less than the bulk threshold. Thus, the waveguides can control the nonlinear effects.
In addition, it can be expected that a balance between dispersion and nonlinear spectrum broadening can be achieved by fabricating particular waveguide arrays which may lead to new high-order asymmetric temporal solitons. These high-order solitons may be stretched and compressed asymmetrically in temporal domain.
Acknowledgments
We acknowledge the financial supports by the National Basic Research Program of China (2007CB307002&2010CB933801), the 111 Project (B07013), the National Natural Science Foundation of China (10604033), SRF for ROCS (SEM), Key International S&T Cooperation Project (2005DFA10170), and Tianjin Natural Science Foundation (09JCYBJC15100).
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